Proceedings of the Japan Academy, Series A, Mathematical Sciences

Deformations of the discrete Heisenberg group

Severin Barmeier

Full-text: Open access


We study deformations of the discrete Heisenberg group acting properly discontinuously on the Heisenberg group from the left and right and obtain a complete description of the deformation space.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 89, Number 4 (2013), 55-59.

First available in Project Euclid: 2 April 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32G08: Deformations of fiber bundles 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx] 22F30: Homogeneous spaces {For general actions on manifolds or preserving geometrical structures, see 57M60, 57Sxx; for discrete subgroups of Lie groups, see especially 22E40} 57S30: Discontinuous groups of transformations

Deformation discrete group properly discontinuous action homogeneous space Heisenberg group


Barmeier, Severin. Deformations of the discrete Heisenberg group. Proc. Japan Acad. Ser. A Math. Sci. 89 (2013), no. 4, 55--59. doi:10.3792/pjaa.89.55.

Export citation


  • A. Baklouti, I. Kédim and T. Yoshino, On the deformation space of Clifford–Klein forms of Heisenberg groups, Int. Math. Res. Not. IMRN 2008, no. 16, Art. ID rnn066, 35 pp.
  • W. M. Goldman, Nonstandard Lorentz space forms, J. Differential Geom. 21 (1985), no. 2, 301–308.
  • T. Kobayashi, Proper action on a homogeneous space of reductive type, Math. Ann. 285 (1989), no. 2, 249–263.
  • T. Kobayashi, Discontinuous groups acting on homogeneous spaces of reductive type, in Representation theory of Lie groups and Lie algebras (Fuji-Kawaguchiko, 1990), 59–75, World Sci. Publ., River Edge, NJ, 1992.
  • T. Kobayashi, On discontinuous groups acting on homogeneous spaces with noncompact isotropy subgroups, J. Geom. Phys. 12 (1993), no. 2, 133–144.
  • T. Kobayashi, Deformation of compact Clifford–Klein forms of indefinite-Riemannian homogeneous manifolds, Math. Ann. 310 (1998), no. 3, 395–409.
  • T. Kobayashi, Discontinuous groups for non-Riemannian homogeneous spaces, in Mathematics unlimited–-2001 and beyond, 723–747, Springer, Berlin, 2001.
  • T. Kobayashi and S. Nasrin, Deformation of properly discontinuous actions of $\mathbf{Z}^{k}$ on $\mathbf{R}^{k+1}$, Internat. J. Math. 17 (2006), no. 10, 1175–1193.
  • R. L. Lipsman, Proper actions and a compactness condition, J. Lie Theory 5 (1995), no. 1, 25–39.
  • S. Nasrin, Criterion of proper actions for 2-step nilpotent Lie groups, Tokyo J. Math. 24 (2001), no. 2, 535–543.
  • A. Weil, Remarks on the cohomology of groups, Ann. of Math. (2) 80 (1964), 149–157.